Question

According to a survey in a​ country, 15 ​% of adults do not own a credit card. Suppose a simple random sample of 200 adults is obtained. Complete parts​ (a) through​ (d) below.

​(a) Describe the sampling distribution of ModifyingAbove p with caret ​, the sample proportion of adults who do not own a credit card. Choose the phrase that best describes the shape of the sampling distribution of ModifyingAbove p with caret below.

A. Approximately normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis greater than or equals 10

B. Approximately normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis less than 10

C. Not normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis less than 10

D. Not normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis greater than or equals 10 Determine the mean of the sampling distribution of ModifyingAbove p with caret . mu Subscript ModifyingAbove p with caret Baseline equals nothing ​(Round to two decimal places as​ needed.)

Determine the standard deviation of the sampling distribution of ModifyingAbove p with caret . sigma Subscript ModifyingAbove p with caret equalsnothing ​(Round to three decimal places as​ needed.)

​(b) What is the probability that in a random sample of 200 ​adults, more than 17 ​% do not own a credit​ card?

The probability is    ? (Round to four decimal places as​ needed.)

Interpret this probability.

If 100 different random samples of __ 200 adults were​ obtained, one would expect ____ to result in more than 17% not owning a credit card.

​(Round to the nearest integer as​ needed.)

​(c) What is the probability that in a random sample of 200 adults, between 12% and 17​% do not own a credit​ card?

The probability is ____? ​(Round to four decimal places as​ needed.)

Interpret this probability.

If 100 different random samples of 200 adults were​ obtained, one would expect ____ to result in between 12% and 17% not owning a credit card.

​(Round to the nearest integer as​ needed.)

​(d) Would it be unusual for a random sample of 200 adults to result in 24 or fewer who do not own a credit​ card? Why? Select the correct choice below and fill in the answer box to complete your choice.

​(Round to four decimal places as​ needed.)

A.The result is not unusual because the probability that ModifyingAbove p with caret is less than or equal to the sample proportion is ___ which is greater than​ 5%.

B.The result isunusual because the probability that ModifyingAbove p with caret is less than or equal to the sample proportion is _____ which is less than​ 5%.

C.The result is notunusual because the probability that ModifyingAbove p with caretis less than or equal to the sample proportion is _____ which is less than​ 5%.

D.The result is unusual because the probability that ModifyingAbove p with caret is less than or equal to the sample proportion is _____ which is greaterthan​ 5%.

Answer 1

a) n has to be less than or equal to 5% of the population,

and np(1-p) has to be greater than or equal to 10.

Evaluate np(1-p)=200*0.15(1-0.15) =25.5

so the distribution is approximately normal because the 2 rules above are valid.

It is approximately normally distributed with mean is 0.15

and standard deviation = .

b) convert 0.17. to a standard normal variable =

the probability that in a random sample of 200 ​adults, more than 17 ​% do not own a credit​ card =P(Z>0.7937) = 1-P(Z<0.7937) = 1-0.7863=0.2137.

If 100 different random samples of 200 adults were​ obtained, one would expect 0.2137*100=21 to result in more than 17% not owning a credit card.

c) convert 0.12. to a standard normal variable =

the probability that in a random sample of 200 adults, between 12% and 17​% do not own a credit​ card = .

If 100 different random samples of 200 adults were​ obtained, one would expect 67 to result in between 12% and 17% not owning a credit card.

d) convert 24 to standard Z score =

Probability that fewer than 24 who do not own a credit card = P(Z<-1.19)=0.1169

which is greater than 0.05

hence option A is right